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We show from a bond valence sum correlation that very high superconducting $T_{c} $ values should be found in optimally hole-doped infinite-layer ACuO$_{2} $ cuprates - up to 160~K for A = Ba. The projected increase in $ T_{c} $across the series arises from "internal pressure" effects as A runs from Mg toBa. We then use density functional theory to investigate these pressure effectson the band structure in an attempt to understand this progressive increase in$ T_{c} $. Where these materials have been synthesised we find good agreementbetween our calculated structural parameters and the experimental ones. We findthat internal pressure associated with increasing ion size does indeed enhancethe superconducting energy gap, as observed, via modifications to theelectronic dispersion. Furthermore, in our calculations, pressure alters thedispersion independently of how it is applied (internal or external) so thatthe superconducting energy gap correlates with the unit-cell volume and aFermi-surface shape-parameter describing ratio of next-nearest-neighbor tonearest-neighbor hopping integrals. We infer an energy scale for the pairinginteraction of the order of 1~eV, well above the magnetic energy scale.

Possible very high transition temperatures in the infinite-layer ACuO$ _{2} $ cuprate superconductor for A={Mg, Ca, Sr, Ba}: A DFT study